Third-order Paired Explicit Runge-Kutta schemes for stiff systems of equations
Introduction
Explicit methods are widely used for the solution of non-stiff systems of equations. However, a wide range of physical applications require the solution of multi-scale locally stiff systems, such as the Navier-Stokes equations, convection-diffusion equation, shallow water equations, and the Euler equations. Numerical stiffness requires prohibitively small time-steps to maintain stability for these types of locally stiff systems, even though this stiffness may arise in only a limited region of the computational domain, such as boundary layers or reaction zones [1], [2], [3]. Implicit schemes are often used to overcome these limitations. However, implicit approaches are often more expensive in terms of computational cost per time-step and memory requirements. Classical Implicit-Explicit (IMEX) Runge-Kutta methods [4], [5], [6], [7], [8], and particularly the recently developed Accelerated-IMEX schemes [9], have demonstrated particular promise in the simulation of such systems, taking advantage of an implicit method for the stiff parts and an explicit method for the non-stiff parts of a system, respectively. Nevertheless, they are relatively complex to implement, and solution of the implicit region can still be expensive. Hence, although explicit methods have the disadvantage of being conditionally stable, they are often preferred due to their simplicity and low cost per time-step [10]. Therefore, ongoing research has been dedicated to alleviating the stability constraints of explicit methods when applied to stiff systems of equations. The primary focus of these studies has been optimizing a scheme's region of absolute stability, a property of the schemes' stability polynomial, enabling the largest possible time step size per stage. For example, we have seen the application of optimization of Runge-Kutta stability polynomials in the works of Van der Houwen et al. [11], Ruuth et al. [12], [13], Ketcheson et al. [14], Parsani et al. [15] Kubatko et al. [16], and Vermeire et al. [17]. In addition, efforts have been made to optimize stability polynomials for multidimensional element types for high-order unstructured methods [15], [18].
Another approach for alleviating the stability of explicit schemes is multirate time-stepping methods. These methods allow different schemes with different time step sizes to be used in different regions of the domain. For example, Constantinescu and Sandu introduced two second-order multirate methods [19], Seny et al. developed a multirate method [20], and Schlegel et al. introduced a recursive multirate scheme for advection equations [21].
Recently, a new method for locally-stiff systems was proposed, referred to as Paired Explicit Runge-Kutta schemes [22]. The P-ERK formulation consists of a family of schemes whose stability polynomials are optimized for a given spatial discretization. With P-ERK different schemes with different stability properties are used in different regions of the domain to increase the global time-step size and accelerate the solution of the locally stiff systems. Members of a P-ERK family have arbitrarily large numbers of stages, but each can possess a different number of active stages. Only active stages require a right hand side derivative evaluation, and the number of active stages is determined based on local stiffness. Hence, P-ERK schemes require significantly fewer derivative evaluations in non-stiff parts of the domain, while retaining the stability benefits of a high-stage count scheme in stiff regions [22]. In this study, we expand P-ERK schemes, originally formulated with second-order accuracy, to third-order accuracy. We then demonstrate the utility of third-order P-ERK schemes when paired with a Flux Reconstruction (FR) spatial discretization applied to the solution of locally-stiff problems with the compressible Navier-Stokes and Euler equations.
This manuscript is organized into six sections. Section 2 reviews explicit schemes and their stability polynomials; in Section 3, we develop general Butcher tableaus for a new family of third-order P-ERK schemes; Section 4 explores optimization of third-order P-ERK stability polynomials for high order unstructured spatial discretization using Von Neumann (Fourier) analysis; in Section 5, we present verification and validation of these new schemes for benchmark test cases, and we evaluate their performance relative to classical explicit RK methods; finally in Section 6, we present conclusions and recommendations for future work.
Section snippets
Explicit Runge-Kutta methods
Runge-Kutta methods are a family of temporal discretizations that use a linear combination of first-order Euler stages to discretize systems of equations of the form [10]
They are typically defined via a Butcher tableau of the form [10] where and s is number of stages.
It has been shown that to achieve at least first-order accuracy, the following condition must be satisfied where the c indicates the positions, within the
Formulation of third-order Paired Explicit Runge-Kutta schemes
In this section, we introduce a formulation for third-order accurate P-ERK schemes. This formulation is developed for an arbitrarily large number of stages, such that multiple families of schemes can be defined. The general form of a family of third-order P-ERK scheme possesses s stages () and e active stages () for each member of the family, denoted by P-ERK. Members with relatively high values of e are employed in stiff regions to enhance stability, and members with relatively
Flux reconstruction
Optimal stability polynomials can be generated for a given set of semi-discrete eigenvalues, , of a spatial discretization. In this study we focus on the high-order Discontinuous Galerkin method obtained via the Flux Reconstruction (FR) approach. The FR approach was first formulated by Huynh [28], and expanded by Wang et al. [29], William et al. [30], and Vincent et al. [31], [32]. FR approach is described by considering a general hyperbolic conservation law of the form where
Verification
To verify the accuracy of the P-ERK family of schemes, we consider an isentropic vortex using the compressible Euler equations. This test case uses a fine mesh and high solution polynomial degree to minimize spatial error, allowing us to isolate and assess the convergence rate of the temporal discretization for verification purposes. The test cases in the following sections will demonstrate how P-ERK schemes can be applied to more practical applications, where spatial error is not strictly
Conclusions
In this study, we have proposed a new family of third-order P-ERK schemes for locally-stiff systems of equations. Building on the original second-order P-ERK formulation, these third-order schemes allow Runge-Kutta schemes with different numbers of active stages to be assigned based on local stiffness criteria, while seamlessly pairing at their interface. We then generated families of schemes optimized for the high-order flux reconstruction spatial discretization, and applied them to a range of
CRediT authorship contribution statement
Siavash Hedayati Nasab: Methodology, Formal Analysis, Generating the new schemes, Validation, Visualization, Writing- Original Draft.
Brian Vermeire: Conceptualization, Software, Resources, Writing - Review and Edit, Supervision, Funding acquisition
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [RGPAS-2017-507988, RGPIN-2017-06773]. This research was enabled in part by support provided by Calcul Quebec (www.calculquebec.ca), WestGrid (www.westgrid.ca), SciNet (www.scinethpc.ca), and Compute Canada (www.computecanada.ca) via a Resources for Research Groups allocation.
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